Hey guys im preparing for a math contest and would like to understand how to do these questions. if anyone has any ideas please let me know.
thanks.
to #16.
There are 2 different ways:
1. Use implicite differentiation:
$\displaystyle x^2+y^2=25~\implies~2x+2y\cdot y'=0$
Plug in the coordinates of the tangent point and solve for y':
$\displaystyle 8-6\cdot y'=0~\implies~y'=\dfrac43$
Therefore the equation of the tangent is
$\displaystyle y+3=\dfrac43(x-4)~\implies~\boxed{y=\dfrac43x-\dfrac{25}3}$
2. $\displaystyle x^2+y^2=25$ is the equation of a circle with the center at the origin and with a radius of 5. The direction from the center to the tangent point is:
$\displaystyle m=\dfrac{-3}4$ The direction of the tangent must be perpendicular to this direction and therefore
$\displaystyle m_t=-\dfrac1{-\frac34} = \dfrac43$
Use the point-slope-formula of a straight line to get the equation of the tangent.